Machine Learning of Time Series Using Time-delay Embedding and Precision Annealing

Tasking machine learning to predict segments of a time series requires estimating the parameters of a ML model with input/output pairs from the time series. Using the equivalence between statistical data assimilation and supervised machine learning, we revisit this task. The training method for the machine utilizes a precision annealing approach to identifying the global minimum of the action (-log[P]). In this way we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series s(tn); tn = t0 + n Delta t and using methods of nonlinear time series analysis show how to produce a DE > 1 dimensional time delay embedding space in which the time series has no false neighbors as does the observed s(tn) time series. In that DE-dimensional space we explore the use of feed forward multi-layer perceptrons as network models operating on DE-dimensional input and producing DE-dimensional outputs.